I think the RA and Dec of a celestial object should change (ever so slightly) with the Earth's position in its orbit. So do we ignore this in general?
This is called "parallax". The observed position of a star (as measured by RA and Dec) does vary ever so slightly.
For example, Proxima centauri, the closest star has a parallax of 0.77 arcseconds. For context, there are 3600 arcseconds in one degree. The moon is about 1800 arcseconds across.
The parallax can only be measured by very careful telescopic observations.
Other stars are further away, and have even smaller parallax.
The underlying RA and Dec grid stays fixed, but stars (in particular nearby stars) move relative to it.